## Thursday, February 13, 2014

### An Olympic Puzzle

February 2014. Russia, Sochi.
Taxi drivers in the city are very busy bouncing all the athletes and spectators in-between the Olympic village and the city. This is a puzzle about two taxis: say Green and Yellow. Green taxi starts from the village and goes toward the city exactly at the same time when Yellow taxi leaves the city in the direction of the village.  They drive toward each other and meet (for the first time) 5 km from the village. Each delivers its passengers and immediately returns back with the new riders: Green taxi now drives from the city to the village while Yellow taxi returns from the village to the city. They meet (for the second time) 3 km from the city.
The question is very simple - how far is the Olympic village from the city?
Here is the sketch created by Kimberly Rose that should help you.

Note that this puzzle is fictional and Googling the real distance will not provide the answer. The puzzle is adapted from a summer boat puzzle from my favorite book of puzzles:

Your thoughts and suggestions are accepted any time until midnight Eastern Time on Sunday, on our Family Puzzle Marathon

kj said...

12 km between the village and the city

We make the assumptions that the taxis drive at constant speeds, and that no time is lost in dropoff/pickup and turnaround.

We have several unknowns:
r_g = the speed of the green taxi
r_y = the speed of the yellow taxi
t_1 = the time to the first meeting
t_2 = the time to the second meeting
D = the distance between the village and the city

we then have these equations:
r_g * t_1 = 5km
r_y * t_1 = (D - 5km)
r_g * t_2 = (D - 5km + 3km) = D - 2km
r_y * t_2 = (5km + D - 3km) = D + 2km

Combine the first and second equations, and the third and fourth equations:
(r_g + r_y) * t_1 = D
(r_g + r_y) * t_2 = (D - 2km + D + 2km) = 2 * D
sub in D from above
(r_g + r_y) * t_2 = 2 * (r_g + r_y) * t_1,
thus t_2 = 2 * t_1
subbing back into the third equation we have
r_g * 2 * t_1 = D - 2km
the left hand side is twice the first equation, so we have
2 * 5km = D - 2km
10km = D - 2km
D = 12km

To check we can sub into the fourth equation:
r_y * 2 * t_1 = D + 2km
and the left hand side is twice the second equation
2 * (D – 5km) = D + 2km
2 * D – 10km = D + 2km
D = 12km

A diagram also helps to see what is going on:

In the first time period, what we call t_1, the taxis cover distance D. In the second time period, t_2, they cover distance 2 * D, and with our assumptions, this time period must be twice the first, so t_2 = 2 * t_1.

Jerome said...

The key to the problem is to observe that the times of both meethings are the same for each vehicle. They have to be or these two vehicles would never meet.

First Meeting
==========
So the first meeting takes place 5 km from the village. The taxi going away from the villiage travels 5 km at a speed of vi

The taxi moving towards the villiage travels d - 5 km at a speed of v2. The time of meeting is the same as stated so the equation is

5/v1 = (d - 5)/v2 Now express this equation in terms of v1/v2
5/(d - 5) = v1/v2.

The second meeting
===============
The second meeting is much, much more complicated.

The taxi going initially towards the city from the villiage has travelled the distance it has gone taking a time of (d+3)/v1

The taxi going initially towards the village from the city has travelled the distance it has gone taking a time of (2d-3)/v2
Equating these two and solving for v1/v2 you get

(d + 3)/(2d - 3) = v1/v2

Two equations Two conditions
=====================
Equate the two meetings since v1 and v2 have not changed (I hope).

5/(d - 5) = (d + 3)/(2d - 3)

10d - 15 = (d - 5)(d + 3)
10d - 15 = d^2 - 2d - 15

From which d = 12 and 0. Zero is meaningless.

Anonymous said...

12 kms

Lulu

Maria said...

This is an interestibg puzzle. It seems incomplete at first, too complex on a second glance ( in fact i somehow ended up with a quadratic equation after which i abandoned it). However it is surprisingly simple once you draw the story, then step back and find the thread to pull everythibg apart.

All we need to notice is that both drivers together cover distance S between the village and the city begore their first meeting, and that together they cover 3 times this distance before their second meeting.

This means that if Green cab driver made 5km before the first meeting, he should have drove 3 times this before the second meeting. 3* 5 is 15 km.
Before the second meeting he covered distance S reachibg the city and then drove another 3km before meetibg the Yellow cab again:

S+3=15
S=12km

Cool, isnt it?
It took me some time to get to it but even my 13year old son found this solution captivating.

Kj sent me a great sketch that i will add to this puzzle as soon as i return back home from the road in a few days.

Bravo to everyone who dared to take this puzzle on! Stay warm.

Jerome said...

Wow!!!!
You are of course kidding??? You couldn't be. It's not April 1st. That is a great problem with a very subtle solution. Every so often one runs across something that is really neat. This sure is an example.